Decentralized Synchronization of an Uncertain Complex Dynamical Network Wei-Song Zhong, Georgi M. Dimirovski and Jun Zhao Abstract— A class of uncertain time-varying complex dynam- ical networks is studied in this paper and a model introduced. On the grounds of that model then the locally and globally decentralized synchronization of these networks are thoroughly investigated. Several network synchronization criteria are de- duced. Especially, the assumptions adopted and decentralized control laws designed are considerably simple. An illustrative example along with the respective numerical and computer simulation results is also given to demonstrate the effectiveness of the proposed synchronization control synthesis. I. INTRODUCTION Analysis and control of complex networks, which repre- sent a systemic structure of a large set of interconnected dynamical nodes, have become a hot topic of great interest in recent years. Two main reasons may well explain this situation status at present. The one emanates from the fact that now science is confronted not with a single complex system, but rather a network of complex systems connected as a large-scale ensemble, following the generic change and expansion of communication and transportation technologies. The other is due to the dramatic change of the everyday relationship between people and complex networks caused by these developments. It has been found already that the topology of a network often affects its functioning. Thus not only the dynamics of each individual node in the network should be considered, but also the topological connectivity of a network if we are to investigate and understand better dynamical behaviors of various complex networks. Traditionally, a network of complex topology is described by a completely random graph, the so-called E-R model, due to famous discoveries of Paul Erdos and Alfred Renyi [1]. More recently, Watts and Strogatz introduced the concept of so-called small-world network [2]-[4], which demonstrates the transition from a regular network to a random network since many real- world complex networks are neither completely regular nor completely random. Another significant discovery in the field of complex networks is the observation that a number of complex networks are scale-free. Small-world phenomenon This work was supported in part by the NSF of P. R. of China, under grand 60574013 and by by Dogus University Fund for Science Wei-Song Zhong and Jun Zhao are with Northeastern University, Key Laboratory Integrated Automation of Process Industry - Ministry of Ed- ucation and School of Information Science and Engineering, respectively, Shenyang, Liaoning, 110004, P.R. of China (e-mails: zwssir@sohu.com; zdongbo@pub.ln.cninfo.net) Georgi M. Dimirovski is with Dogus University, Dept. of Computer Engineering, Acibadem, Kadikoy, TR-34722, Istanbul, Rep. of Turkey, and with SS Cyril and Methodius University, Faculty of Electrical Eng. Info. Technologies, MK-1000 Skopje, Rep. of Macedonia (e-mail: gdimirovski@ dogus.edu.tr) and scale-free feature have been shown to play critical roles in complexity [4]. More recently, a general scale-free dynamical network model was discussed in [18] first bringing in the significant result that the synchronizability of a scale-free dynamical network is robust against random removal of nodes yet it is fragile to a specific removal of the most highly connected nodes. Synchronization and dynamical behaviors in complex networks were studied further on the grounds of that model [7], [8], [10], [12], [17], [18], [19]. In particular, the uniform and non-uniform pinning control strategy, including specifi- cally pinning scheme and randomly pinning scheme, is used to stabilize scale-free networks [6], [11], [14], [15], [16]. In [5], a general time-varying complex dynamical network model was presented and also the synchronization problem was studied. In these investigations, an essential requirement is that the structure of the network and the coupling functions are known a priori. Works [9] and [13] proposed an uncertain dynamical network model with an unknown but bounded nonlinear function and discussed its robust adaptive synchronization. Paper [19] has provided results on the transition to chaos in complex dynamical networks while [20] used hybrid controls to synchronize chaotic systems. Decentralized robust controls of network-like large-scale nonlinear systems were studied in [21]-[26]. All the results derived in those papers were proven to be effective for the specific large-scale system investigated. In this paper, a controlled time-varying complex dy- namical network model is proposed. Further, by exploring the combined application of Lyapunov stability theory and nonlinear robust decentralized control method, the present paper shows some robust decentralized controllers can indeed be designed for the same task as described in [9] and [13]. It shows that the proposed synthesis can ensure the states of the dynamical network locally and globally asymptotically synchronize with an arbitrarily assigned state of an isolate node in the network. Also, the synthesis for the controlled time-varying complex dynamical network model is shown to be significantly simplified using the similarity structure. This paper is organized as follows. An uncertain time- varying complex dynamical network model is presented and some preliminaries are introduced in Section II. In Section III, the locally and globally decentralized synchronization approaches of the network are studied. Section IV presents an application example along with the respective numeri- cal and simulation results to verify the theoretical results and demonstrate the effectiveness of the proposed control method. Conclusion and references follow thereafter Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 WeB21.6 1-4244-0989-6/07/$25.00 ©2007 IEEE. 1437